132 



PRACTICAL MATHEMATICS 



Equation of the tangent is y = mx + c, or y = x l-38I8e + c 

 but 0-84,l5e = 1-3818* + c 



and c = - 0-5403* 



and y = 1-3818*0 - 0-5403* 



= 3-7580 - 1-469 

 i 



Equation of the normal is w = x+ c-, or w = , * n + c, 



m 1-3818* 



but 0-8415* = - 



and 



1-3818* 

 2-554 



x 



+ 2-554 



= -0.26630 + 2-554. 



81. The Angle between a Line and a Curve at their Point of 

 Intersection. In this case the required angle is the angle between 

 the line and the tangent to the curve at the point of intersection. 



FIG. 37- 



Let ABC (Fig. 37) be the curve and KL the line. C is the point 

 of intersection, and PC is the tangent to the curve at the point C. 



Then the required angle is PCL. 



If y = f(x) be the equation to the curve, and y = mx + c the 

 equation of the line, the co-ordinates of the point C are the values 

 of x and y which satisfy these two equations simultaneously. 



Let these values be x = h and y = k. 



Then the slope of the tangent (?-] 



\dx/x=h 



and hence 



. 



This gives the inclination of the tangent PC 



